One is Jupiter's moon Europa and another is Saturn's moon Titan. Both of them have the problem of having at low temperature as Sun's heat cannot reach them.

Jupiter is made up mostly of Hydrogen so is our Sun. Can we ignite Jupiter so that it will produce enough heat to warm these two earth like planets/satellites? What can be consequences of any of such attempt?

Even if you had some artificial means to heat it up and get "ignition" it couldn't be sustained. Basically the gravity wouldn't be able to contain the higher pressures, and the planet would expand and cool down. This is the same dynamics that keep stars stable. This is very similar to the problem of scaling down flamesize, if you get the scale too small, then the thermal loses exceed the thermal generation and it cools down and dies out. In 2010 the monoliths contained some sort of magic fusion reactor technology!
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Omega CentauriMar 3 '11 at 22:45

There is a minor change to yyahn's answer. The isotopes Deuterium, and Lithium7 which are present in small amounts -left over from the big bang, can fuse at lower mass than pure hydrogen burning. The estimate is at around 65 mass Jupiter Lithium fuses with hydrogen to form two helium nuclei. Brown dwarfs as light as 13 mass Jupiter can ignite Deuterium fusion. So we
seem to have some weak forms of fusion (on low abundance fuels) that can happen for some range of brown dwarf masses. See wikipedia on brown dwarfs.

this means it won't self-ignite, but if a thermonuclear device were set off, would it burn? This is a question of areal density I think ($\rho R$)
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JeremyDec 23 '10 at 20:59

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@Jeremy - no, it wouldn't. Fusion produces enough heat that it causes a star to expand which typically causes fusion to turn off because the density gets too low. Jupiter wouldn't be able to stably fuse hydrogen - it would quickly go out.
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spencer nelsonFeb 21 '11 at 3:30

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@Spencer, you are correct in the limit of hydrodynamic equilibrium. But if the burn wave propagation is faster than the dynamical time, it would ignite. This is what happens in type 1a supernovae.
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JeremyApr 11 '11 at 15:49

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@Jeremy: Also, if by "burn" you mean a chemical reaction like a flame, you would need free oxygen for that, and there is none. It's also worth noting that we've observed comet hits on Jupiter, which produced explosions similar to those produced by thermonuclear bombs (actually, releasing more energy on the order of many thousand times anything we mere mortals can produce). Consider that this happens on a regular basis, and has already happened thousands or millions of times.
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ErnieMay 17 '11 at 21:06

A star is not burning in the typical sense of burning that we have on earth. It's fusing hydrogen into helium. And it does this because it has so much mass that gravity is trying to collapse the star but the immense pressure means fusion takes place (creating an outward pressure to counteract the gravity pulling inwards creating a balance which must be maintained to keep the sun from collapsing inwards or exploding. ) so a star is defined by it's mass. You have brown dwarfs that are almost massive enough to start fusion.

This idea seems to crop up now and then. I will have to confess the thought occurred to me as a kid in grade school. The AC Clarke “2010” featured Jupiter being imploded by black monoliths and turning into a star.
Even if you get a fusion bomb into the core of Jupiter, which would be tough to do, the explosion would have no measurable effects on the surface. There simply is insufficient pressure to fuse nuclei together. You would need some sort of pressure producing system that is also stable. You might consider putting a small black hole (an Earth mass BH or so) in the core of Jupiter, which might create a zone of imploding material that has nuclear fusion. The rub with this idea is you also ultimately turn Jupiter into a 3-4 meter radius black hole. A collection of Hawking radiating black holes might be better, but their masses would have to be tuned so they don’t absorb or lose mass. This would mean turning Jupiter into a Hawking radiation source as a "star." That is not a stable system. So to play it safe you tune the Hawking BHs to radiate away and then replenish them later. Again, none of this is a small feat.

Of course the idea is to create a sort of mini-solar system with the various moons. So even if you could by some means turn Jupiter on as a star the results after a few thousand years would be to turn the Jovian and Saturnian moons into liquid bodies. These moons would not turn into Earth-like planets.

Another option would be to suspend fusion "lanterns" in Jupiter or Saturn's atmosphere and guide the output toward a target moon. You could also put a soletta, a series of lightweight mirrors acting as a lens, in between the moon and the sun. The multi-body orbital dynamics and the system of mirrors (or other "thrusters") to keep it from being "blown away" by the solar wind gets pretty complicated though.

He meant for two types of ignitions. One is fusion and another is burning. Fusion is not possible because it don't have enough mass of hydrogen. The second is also not possible because the absence of oxygen.

The chemical equation for the combustion of hydrogen gas is
$2H_2+O_2 \rightarrow 2H_2 O$

The atmosphere of Jupiter is mostly helium and hydrogen. We can burn Jupiter's hydrogen if we have enough oxygen. But Jupiter is very big so we need a LOT of oxygen to burn a significant amount Jupiter's hydrogen (relative to Jupiter's volume).

Further to Lawrence B. Crowell's excellent answer above: in the past, Jupiter was often described (temptingly but mistakenly) as a failed star. It's not even close to the mass needed for self-ignited fusion, and dropping e.g. an Earth-mass mini black hole wouldn't light or sustain nuclear fusion. The mass would still be too low.

However, supposing you could corral a mini black hole and nudge it close enough for Jupiter to capture and swallow it, the black hole would generate huge amounts of energy by consuming anything and everything that dropped into it, but it would also generate a tremendous counterpressure that would prevent its swallowing Jupiter at a gulp.

After oscillating countless times and who-knows-how-many years, the black hole would settle in the center and eventually (inevitably) suck nearly all of Jupiter into itself, all the while producing gobs and gobs of energy that might or might not be useful to anyone. Because of the counterpressure, the process would take a very (I mean a very) long time, but it wouldn't be fusion. For that you'd need enough mass for at least a dwarf star, whether brown, white, or lavender.

[Edit/afterthought]: Only two candidate Earths? Have your forgotten dear old Mars? Incidentally, Jupiter's existing magnetic field pretty much rules out the possibility of ever establishing a human presence on Jupiter's biggest, most succulent moons. Alas for Robert A. Heinlein's 'Farmer in the Sky'...

The most likely answer is that probably Jupiter is already ignited: it emits lots of infrared radiation that as far as i know, its largely unexplained. It just doesn't have enough mass to radiate more energy than a extremely dim brown dwarf, so it naively looks to us as a planet